Integrand size = 15, antiderivative size = 75 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {-a B+A c x}{4 a c \left (a+c x^2\right )^2}+\frac {3 A x}{8 a^2 \left (a+c x^2\right )}+\frac {3 A \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \] Output:
1/4*(A*c*x-B*a)/a/c/(c*x^2+a)^2+3/8*A*x/a^2/(c*x^2+a)+3/8*A*arctan(c^(1/2) *x/a^(1/2))/a^(5/2)/c^(1/2)
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {\sqrt {a} \left (-2 a^2 B+5 a A c x+3 A c^2 x^3\right )}{\left (a+c x^2\right )^2}+3 A \sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c} \] Input:
Integrate[(A + B*x)/(a + c*x^2)^3,x]
Output:
((Sqrt[a]*(-2*a^2*B + 5*a*A*c*x + 3*A*c^2*x^3))/(a + c*x^2)^2 + 3*A*Sqrt[c ]*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c)
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {454, 215, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 454 |
\(\displaystyle \frac {3 A \int \frac {1}{\left (c x^2+a\right )^2}dx}{4 a}-\frac {a B-A c x}{4 a c \left (a+c x^2\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {3 A \left (\frac {\int \frac {1}{c x^2+a}dx}{2 a}+\frac {x}{2 a \left (a+c x^2\right )}\right )}{4 a}-\frac {a B-A c x}{4 a c \left (a+c x^2\right )^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3 A \left (\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {x}{2 a \left (a+c x^2\right )}\right )}{4 a}-\frac {a B-A c x}{4 a c \left (a+c x^2\right )^2}\) |
Input:
Int[(A + B*x)/(a + c*x^2)^3,x]
Output:
-1/4*(a*B - A*c*x)/(a*c*(a + c*x^2)^2) + (3*A*(x/(2*a*(a + c*x^2)) + ArcTa n[(Sqrt[c]*x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[c])))/(4*a)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Time = 0.85 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {2 A c x -2 B a}{8 a c \left (c \,x^{2}+a \right )^{2}}+\frac {3 A \left (\frac {x}{2 a \left (c \,x^{2}+a \right )}+\frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 a \sqrt {a c}}\right )}{4 a}\) | \(70\) |
risch | \(\frac {\frac {3 A c \,x^{3}}{8 a^{2}}+\frac {5 A x}{8 a}-\frac {B}{4 c}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 A \ln \left (c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}+\frac {3 A \ln \left (-c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}\) | \(83\) |
Input:
int((B*x+A)/(c*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/8*(2*A*c*x-2*B*a)/a/c/(c*x^2+a)^2+3/4*A/a*(1/2*x/a/(c*x^2+a)+1/2/a/(a*c) ^(1/2)*arctan(c*x/(a*c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.83 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\left [\frac {6 \, A a c^{2} x^{3} + 10 \, A a^{2} c x - 4 \, B a^{3} - 3 \, {\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{16 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac {3 \, A a c^{2} x^{3} + 5 \, A a^{2} c x - 2 \, B a^{3} + 3 \, {\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{8 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \] Input:
integrate((B*x+A)/(c*x^2+a)^3,x, algorithm="fricas")
Output:
[1/16*(6*A*a*c^2*x^3 + 10*A*a^2*c*x - 4*B*a^3 - 3*(A*c^2*x^4 + 2*A*a*c*x^2 + A*a^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)))/(a^3*c ^3*x^4 + 2*a^4*c^2*x^2 + a^5*c), 1/8*(3*A*a*c^2*x^3 + 5*A*a^2*c*x - 2*B*a^ 3 + 3*(A*c^2*x^4 + 2*A*a*c*x^2 + A*a^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a))/( a^3*c^3*x^4 + 2*a^4*c^2*x^2 + a^5*c)]
Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.65 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=A \left (- \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16}\right ) + \frac {5 A a c x + 3 A c^{2} x^{3} - 2 B a^{2}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \] Input:
integrate((B*x+A)/(c*x**2+a)**3,x)
Output:
A*(-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x)/16 + 3*sqrt(-1/(a **5*c))*log(a**3*sqrt(-1/(a**5*c)) + x)/16) + (5*A*a*c*x + 3*A*c**2*x**3 - 2*B*a**2)/(8*a**4*c + 16*a**3*c**2*x**2 + 8*a**2*c**3*x**4)
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}}{8 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {3 \, A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} \] Input:
integrate((B*x+A)/(c*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*(3*A*c^2*x^3 + 5*A*a*c*x - 2*B*a^2)/(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4 *c) + 3/8*A*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2)
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, A \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} + \frac {3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \] Input:
integrate((B*x+A)/(c*x^2+a)^3,x, algorithm="giac")
Output:
3/8*A*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/8*(3*A*c^2*x^3 + 5*A*a*c*x - 2*B*a^2)/((c*x^2 + a)^2*a^2*c)
Time = 6.68 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {5\,A\,x}{8\,a}-\frac {B}{4\,c}+\frac {3\,A\,c\,x^3}{8\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {3\,A\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}} \] Input:
int((A + B*x)/(a + c*x^2)^3,x)
Output:
((5*A*x)/(8*a) - B/(4*c) + (3*A*c*x^3)/(8*a^2))/(a^2 + c^2*x^4 + 2*a*c*x^2 ) + (3*A*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(5/2)*c^(1/2))
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.59 \[ \int \frac {A+B x}{\left (a+c x^2\right )^3} \, dx=\frac {3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a^{2}+6 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) a c \,x^{2}+3 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}\, \sqrt {a}}\right ) c^{2} x^{4}-2 a^{2} b +5 a^{2} c x +3 a \,c^{2} x^{3}}{8 a^{2} c \left (c^{2} x^{4}+2 a c \,x^{2}+a^{2}\right )} \] Input:
int((B*x+A)/(c*x^2+a)^3,x)
Output:
(3*sqrt(c)*sqrt(a)*atan((c*x)/(sqrt(c)*sqrt(a)))*a**2 + 6*sqrt(c)*sqrt(a)* atan((c*x)/(sqrt(c)*sqrt(a)))*a*c*x**2 + 3*sqrt(c)*sqrt(a)*atan((c*x)/(sqr t(c)*sqrt(a)))*c**2*x**4 - 2*a**2*b + 5*a**2*c*x + 3*a*c**2*x**3)/(8*a**2* c*(a**2 + 2*a*c*x**2 + c**2*x**4))